is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
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17
Yes.
Rudin is using the convention that $0^0 = 1$.
There are a lot of arguments in favour of this convention, but really in this case it just comes down to: "if I use the convention that $0^0 = 1$, then all my formulas hold well, whereas if I use a different convention, then I'll have to separate the $n=0$ term from the rest of the sum everywhere throughout my book, resulting in heavier, harder to read formulas."
Also note the two related conventions that Rudin might also be using: \begin{align*} 0! & = 1 \\ \prod_{n \in \emptyset} c_n & = 1 \end{align*}
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1$\begingroup$ @Mathematicsenjoyer What's not rigorous? Defining a convenient notation? $\endgroup$– StefCommented Dec 25, 2023 at 14:09
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1$\begingroup$ @Mathematicsenjoyer Note that it's okay to define whatever notation you want. The one thing you're not allowed to do is use the same notation for two different things and draw abusive conclusions by equaling the two things just because they have the same notation. $\endgroup$– StefCommented Dec 25, 2023 at 14:13
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1$\begingroup$ @Mathematicsenjoyer As an example, remember how at some point a math teacher defined $a^n = a \times a \times ... \times a$ for any real $a$ and positive integer $n$, and then later another math teacher defined the exponential function $\exp(x) = \sum x^k / k!$, and then the teacher wants to define the notation $e^x = \exp(x)$. But at this point $e^n$ would be an ambiguous notation: it might mean either $e \times ... \times e$, or $\exp(n)$. So first, the teacher has to prove that $\exp(n) = e \times ... \times e$ when $n$ is integer; and then it's okay. $\endgroup$– StefCommented Dec 25, 2023 at 14:16
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1$\begingroup$ @Mathematicsenjoyer So here it's the same thing; Rudin defines $0^0$ as a notation which is equal to $1$; and before choosing this convention, he made sure that it wouldn't lead him to draw abusive conclusions. $\endgroup$– StefCommented Dec 25, 2023 at 14:17
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1$\begingroup$ @Mathematicsenjoyer No. $\frac 0 0$ does not make sense; I'm not aware of any mathematician who ever used the weird, confusing, useless and error-prone convention that $\frac 0 0 = 1$. However $0^0 = 1$ is used a lot and there are good reasons for that. In fact there is more that one question about it on this website, with insightful answers as to why it is a good convention, and also with mentions about the (small) limitations and caveats of this notation. $\endgroup$– StefCommented Dec 25, 2023 at 14:20